Introduction

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In order to have a complete theory of gravity, we need to know

• How particles behave in curved spacetime.
• How matter curves spacetime.

The first question is answered by postulating that free particles [ i.e. no force other than gravity ] follow timelike or null geodesics. We will see later that this is equivalent to Newton's law   in the week field limit [ ].

The second requires the analogue  of . We first consider the vacuum case [ ] .

The easiest way to do this is to compare the geodesic deviation  equation derived in the last section with its Newtonian analogue. In Newtonian theory the acceleration of two neighboring particles with position vectors and are:

so the separation evolves according to:

This gives us

since

This clearly is analogous to the geodesic deviation equation

provided we relate the quantities and

Both quantities have two free indices, although the Newtonian index runs from 1 to 3 while in the General Relativity case it runs from 0 to 3.

The Newtonian vacuum equation is which implies that

so we can write

Since is arbitrary we end up with

These are the vacuum field equations .